Random fields beyond the null: building models from critical points

Random field theory (RFT) is has been used in signal detection in the"massively univariate" linear models of neuroimaging.Such analyses preclude building multivariate models of activity, comparing activations in different regions of the brain. The main tool used in such analyses is the Kac-Rice formula that enables a Palm theory for the process conditioning on the peaks in the (massively univariate) statistical maps. We describe how to use this Palm theory and a variant of the LASSO to build multivariate models that allow comparison of different peaks in the image as well as quantify uncertainty about their spatial location. The neuroimaging application is a particular example of a canonical "bump hunting" problem, for which a similar method of inference is applicable.